input output anaylysis

1

Input-Output Analysis

World Bank

Skopje, October 26-28, 2011

2

Outline Input-Output Definitions

Simplest form of Input-Output Analysis (a quick review)

Input-Output Analysis with Examples

Input-Output Multipliers

Power and Problems of Input-Output Analysis

Appendix 1: Mathematical Exercises on IO Analysis with Matrices

Appendix 2: Derivation of IO Multipliers

3

PART I:

Input-Output Definitions

4


“The assessment of change in overall economic activity as

the result of some corresponding change in one or several

activities”.

“An economic analysis, in which the interdependence of an

economy's various productive sectors is observed by viewing

the product of each industry both as a commodity demanded

for final consumption and as a factor in the production of

itself and other goods” (Encyclopedia Brittanica)

5


“A methodology for investigating production relations

among primary factors, intersectoral flows, final demands,

and transfers.”

“Input-output analysis considers inter-industry relations in

an economy, depicting how the output of one industry goes

to another industry where it serves as an input, and thereby

makes one industry dependent on another both as customer

of output and as supplier of inputs.“ (Wikipedia)

6

Wassily Leontief (1905 - 1999)

• The “Structure of the American

Economy”, 1919-1939 (1941)

• Input-Output Economics (1966)

Nobel Prize (1973)

7 7

Wassily Leontief

• Born in St. Petersburg, 1906

• Got Ph.D. in Econ in Berlin

• Moved to NYC in 1931

• Joined Harvard econ faculty in 1932

• Constructed first input-output tables of US

• Questioned the H-O Theory after WWII

• 1973 Nobel Prize

8 8

Leontief Paradox

• Leontief reached a paradoxical conclusion that

the US --the most capital abundant country in

the world by any criterion-- exported labor-

intensive commodities and imported capital-

intensive commodities in 1947.

• This result has come to be known as the Leontief

Paradox.

9 9

Input-Output Tables

• Leontief took the profession by surprise and

stimulated an enormous amount of empirical and

theoretical research on the subject.

• To perform the test, Leontief used the 1947 input-

output table of the US economy.

10 10

Paradox continued

• He aggregated industries into 50 sectors, but only 38

industries produced commodities that enter the international

markets, and the remaining 12 sectors were created for

accounting identities and non-traded goods.

• He also aggregated factors into two categories, labor and

capital. He then estimated the capital and labor

requirements to produce:

• One million dollars' worth of typical exportable and

importable in 1947.

11 11

Paradox continued

• The US seems to have been endowed with

more capital/worker than any other country in the world in 1947.

• Thus, the HO theory predicts that the US exports would have required more capital per worker than US imports.

• However, Leontief was surprised to discover that US imports were 30% more capital-intensive than US exports.

12 12

Paradox continued…

• At first, Leontief was criticized on statistical grounds.

• Swerling complained that 1947 was not a typical year:

the postwar disorganization of production overseas was

not corrected by that time.

• In 1956 Leontief repeated the test for US imports and

exports which prevailed in 1951. In his second study,

Leontief aggregated industries into 192 industries. He

found that US imports were still more capital-intensive

than US exports.

• US imports were 6% more capital-intensive.

13 13

Explanations

• Leontief himself suggested an explanation for

his own paradox. He argued that US workers

may be more efficient than foreign workers.

• Perhaps U.S. workers were three times as

effective as foreign workers.

14 14

Higher US Productivity

• It means that the average American worker is

three times as effective as he would be in the

foreign country.

• Given the same K/L ratio, Leontief attributed

the superior efficiency of American labor to

superior economic organization and

economic incentives in the U.S.

15 15

More than two factors

• More recent tests recognize that more than

two types of production factors are relevant.

• Results for H-O are not that disappointing

when we take more factors into consideration.

16 16

H-O works for

some US sectors…

• In general, trade patterns fit the H-O theory

reasonably well but certainly not perfectly.

• The US is relatively abundant in skilled labor, and

tends to be a net exporter of products that are

skilled-labor-intensive or technology-intensive,

including aircraft and medical instruments.

17

• An IO model is centered on the idea of inter-industry

transactions:

– Industries use the products of other industries to produce

their own products.

– For example - automobile producers use steel, glass,

rubber, and plastic products to produce automobiles.

– Outputs from one industry become inputs to another.

– When you buy a car, you affect the demand for glass,

plastic, steel, etc…

Back to the definitions:

Input-Output Model

18

Basic Input-Output Logic

Automobile Factory

Steel Glass

Tires Plastic Other

Components

19

From the Tire

Producer’s

Perspective

Tire Factory

School

Districts

Trucking

Companies

Automobile

Factory

Individual

Consumers

INTER-

MEDIATE

DEMAND

FOR

TIRES

FINAL

DEMAND

FOR TIRES

20


The implicit assumption in economic base techniques is that

each basic sector job has a multiplier (or ripple) effect on

the wider economy because of purchases of non-basic goods

and services to support the basic production activity. (the

basic sector drives the non-basic sector)

However, we know that non-basic sector businesses

purchase non-basic goods and services and basic sector

businesses purchase Basic sector goods and services. There

are inter-industry linkages not contained within a basic

economic model (basically the multiplier model you know

from your intermediate macro course). The economy is

much more complex than the economic base techniques

allow or attempt to model.

21


The central advantage of Input-Output analysis is that it

tries to estimate these inter-industry transactions and use

those figures to estimate the economic impacts of any

changes to the economy.

Instead of assuming a change in a basic sector industry

having a generalized multiplier effect, the IO approach

estimates how many goods and services from other sectors

are needed (inputs) to produce each dollar of output for the

sector in question. Therefore it is possible to do a much

more precise calculation of the economic impacts of a given

change to the economy.

22

IO Conceptualization of the Economy

• The major conceptual step is to divide the economy into

“purchasers” and “suppliers”:

--Primary Suppliers: They sell primary inputs (labor, raw

materials) to other industries. Payments to these suppliers

are “primary inputs” because they generate no further

sales. (example: Households)

--Intermediate Suppliers: They purchase inputs for

processing into outputs they supply to other firms or to

final purchasers. (example: Automaker)

23

IO Conceptualization of the Economy • The major conceptual step is to divide the economy into

“purchasers” and “suppliers”:

--Intermediate Purchasers: They purchase outputs of suppliers

for use as inputs for further processing. (example:

Automaker)

--Final Purchasers: Purchase the outputs of suppliers in their

final form and for final use. (example: Households)

• Intermediate Suppliers and Intermediate Purchasers are the

same thing!

• Primary Suppliers and Final Purchasers may or may not be

the same entities. When they are the same (households),

these activities are understood as separate activities.

24

Simplified Circular Flow View of The Economy

Households buy

the output of

business: final

demand or Yi

Businesses Households

Goods & Services

$$ Consumption Spending (Yi)

Labor

$$ Wages & Salaries

Businesses

Businesses purchase from

other businesses to produce

their own goods / services.

This is intermediate

demand or xij (output of

industry i sold to industry j)

Households sell

labor & other

inputs to business

as inputs to

production

25

PART II:

Input-Output Model for a Simple Economy

26


• Consider a simple economy with two industries:

– a lumber industry

– a power industry

• Suppose that production of 10 units of power require 4 units

of power and 25 units of power require 5 units of lumber.

• 10 units of lumber require 1 unit of lumber and 25 units of

lumber require 5 units of power.

• If surplus of 30 units of lumber and 70 units of power

are desired, how much would be the gross production of

each industry.

27

Creating a Technology Matrix

• First step would be

converting all numbers to

percentages:

• Powerpower: 4/10 = 0.4

• Lumberpower: 5/25 = 0.2

• Lumberlumber: 1/10 = 0.1

• Powerlumber: 5/25 = 0.2

Outputs

Inputs Power Lumber

Power 0.4 0.2

Lumber 0.2 0.1

28

Creating a Technology Matrix

• Now we can use this

information to create the so

called “technology matrix”

or “Leontief matrix”.

Outputs

Inputs Power Lumber

Power 0.4 0.2

Lumber 0.2 0.1

1.2.

2.4.A

29

The Gross Production Matrix

• The gross production matrix for the economy can be

represented by the column matrix:

• Where x1 is the gross production of power and x2 is the

gross production of lumber.

2

1

x

xX

30

The Technological Equation

• X (or IX where I is the identity matrix) is the amount of

production that is desired.

• AX is the amount of actual production.

• So IX-AX=(I-A)X is the amount of surpluses, D, (also

called final demands).

31

The Technology Equation

• Is called the “technology equation”.

DXAI )(

32

Back to our Simple Economy

• Original Question : If surplus of

30 units of lumber and 70 units

of power are desired, find the

gross production of each

industry.

• In other words, what is the gross

production which would satisfy

the final demand D (70,30)?

• Find X

30

70D

33


• To find X, take the inverse of (I-A) [if it

exists]

DAIX

DXAI

1)(

)(

34

9.2.

2.6.

1.2.

2.4.

10

01AI

2.14.010

4.08.101

2.14.010

0667.1333.1

1333.833.0

0667.1333.1

109.2.

0667.1333.1

10

01

9.2.

2.6.

Finding the inverse of I-A:


35

9.2.

2.6.

1.2.

2.4.

10

01AI

2.14.

4.8.1

6.2.

2.9.2

6.2.

2.9.

04.54.

1

6.2.

2.9.

))2.(*)2.(()9.*6(.

1

Finding the inverse of I-A (alternative way):


36


• We found that the inverse of I-A is:

• To find the amount to produce for the desired amount of demand, we must multiply the inverse of I-A and D:

• Hence the gross production are :

– Lumber : 64 units

– Power : 138 units

64

138

30

70

2.14.

4.8.1

)(

2.14.

4.8.1)(

1

1

X

DAIX

AI

37

Input-Output Model for a Simple Economy Another Example (3x3)

• The economy of a hypothetical developing country is based on agricultural products, steel, and coal.

• An input of 1 ton of agricultural products requires an input of 0.1 ton of agricultural products, 0.02 ton of steel, and 0.05 ton of coal.

• An output of 1 ton of steel requires an input of 0.01 ton of agricultural products, 0.13 tons of steel, and 0.18 tons of coal.

• An output of 1 ton of coal requires an input of 0.01 ton of agricultural products, .2 tons of steel, and 0.05 ton of coal.

• Find the necessary gross productions to provide final demands of 2350 tons of agricultural products, 4552 tons of steel, and 911 tons of coal.

• What is the technology matrix?

38

Output

Input Agriculture Steel Coal

Agriculture 0.1 0.01 0.01

Steel 0.02 0.13 0.2

Coal 0.05 0.18 0.05

05.18.05.

2.013.02.

01.01.1.0

A

Input-Output Model for a Simple Economy Another Example (3x3) – Technology Matrix:

39

• What is the matrix of final demands?

• Find the technological equation.

• What is (I-A)-1?

• What is the production matrix?


40

• What is the surplus matrix?

911

4552

2350

D


41

• Find the technological equation:

991

4552

2350

05.18.05.

20.13.02.

01.01.1.0

100

010

001

)(

X

DXAI


42

• What is (I-A)-1?

10.1229.066.

254.20.1041.

015.016.11.1

)( 1AI


http://www.wikihow.com/Inverse-a-3X3-Matrix

http://www.easycalculation.com/matrix/matrix-inverse.php











43

• What is the production matrix?

• Thus in our country to achieve the

desired levels of final demand

2700 units of agriculture, 5800

units of steel, and 2200 units of

coal must be produced.

2200

5800

2700

)( 1 DAIX


46

Closed Leontief Models

• The technological equation for a closed

Leontief model is:

• Where 0 is actually a column matrix of all

zeros.

0)( XAI

47

PART III:

Input-Output Analysis with Examples

48

The Structure of IO Analysis • The ultimate goal of the Input-Output Analysis technique is

to generate a Total Requirements Table that shows the flows

of dollars between industries in the production of output for

a given sector.

• To arrive at this final result, IO Analysis requires two earlier

steps:

1) Transactions table: Contains basic data on the flows of

goods and services among suppliers and purchasers during a

study year.

2) Direct Requirements table: Derived from the transactions

table, this shows the inputs required directly from different

suppliers by each intermediate purchaser for each unit of

output that purchaser produces.

49

The Transactions Table

(in thousands of units)

Intermediate Purchasers Final Purchasers Total

--Agriculture --Manufacturing --Households Sales (outputs)

Intermediate Suppliers

--Agriculture 10 30 60 100

--Manufacturing 5 10 35 50

Primary Suppliers

--Households 85 10 15 110

Total Purchases (inputs) 100 50 110 260

Direct Requirements Table


--Agriculture --Manufacturing

Intermediate Suppliers Every unit of output

--Agriculture 0.10 0.60 requires inputs of a certain

--Manufacturing 0.05 0.20 amount from other areas

Primary Suppliers of the economy.

--Households 0.85 0.20

Total Purchases (inputs) 1.00 1.00

Purchasers

The Transaction Table and Direct Reqs Tables

50



Intermediate Purchasers

--Agriculture --Manu

Intermediate Suppliers

--Agriculture 0.10 0.60

--Manufacturing 0.05 0.20

Primary Suppliers

--Households 0.85 0.20

Total Purchases (inputs) 1.00 1.00

Total Requirements Calculation (First Round)


Sales to Sales as Direct Inputs

Final Purch. To Agr To Manu Total

By Agriculture 200 20 60 80

By Manufacturing 100 10 20 30

By Households 0 170 20 190

Total indirect rounds

By All Supliers 300 300

The First Round of Economic Impacts

To

Rd. 2

51

Total Requirements Calculation (Second Round)




By Agriculture 80 8.0 18.0 26.0

By Manufacturing 30 4.0 6.0 10.0

By Households 0 68.0 6.0 74.0

Total indirect rounds 110.0

Total Requirements Calculation (Third Round)




By Agriculture 26 2.6 6.0 8.6

By Manufacturing 10 1.3 2.0 3.3



Total Requirements Calculation (Fourth Round)




By Agriculture 8.6 0.9 2.0 2.8

By Manufacturing 3.3 0.4 0.7 1.1



The Second-Fourth Rounds of Econ. Impacts

and so on

until the mult.

effect ends

52

Total Direct and Indirect Requirements Calculation


Sales to Final Total Total Total

Purchasers Direct Sales Indirect Sales Sales

Agriculture 200.0 80.0 38.7 318.7

Manufacturing 100.0 30.0 14.9 144.9

Households -- 190.0 109.6 299.6

Total 300.0 300.0 163.1 763.1

The Total Requirements Results

When:

1) there are “Final Sales” of Agriculture = 200 and “Final Sales” of Manufacturing = 100

2) we see a Total Economic Impact = 763.1, with that impact broken down as:

i) 300.0 in Initial Sales to Final Purchasers

ii) 300.0 in Total Direct Sales

ii) 163.1 in Total Indirect Sales

The 300 units in Final Sales generate an additional 463.1 units of economic activity. This illustrates the multiplier effect captured by IO models.

53

Total Requirements Table

Every Unit in Final Demand of…

Requires Total Sales by Agriculture Manufacturing

Agriculture 1.15 0.86

Manufacturing 0.07 1.29

Households 1.00 1.00

Total 2.22 3.15

For Agriculture 1.00 Sales to Final Purchasers

1.00 Sales by Primary Suppliers

0.22 Interindustry transactions

Similar to our Base Multiplier in Econ Base Theory

A 1.0 unit increase in demand for agriculture leads to

a total of 2.22 of sales.

For Manufacturing 1.00 Sales to Final Purchasers

1.00 Sales by Primary Suppliers

1.15 Interindustry transactions

Similar to our Base Multiplier in Econ Base Theory

A 1.0 unit increase in demand for manufacturing leads to

a total of 3.15 of sales.

The Total Requirements Table

54

Case Study: Sugar

How Sweet it is?

55

Case: Sugar industry

• In US, EU, Japan, the domestic price of your

sugar is more than double the world price.

• For the US, the net cost of protectionist policies

is close to $1bn per year.

• The sugar industry is not big, just 60,000 people

or 0.04% of total labor force.

56

Sugar case

• But industry is well organized.

• The big sugar producers in Florida gain $65m per year

from the protectionist policies.

• To defend these profits, they donate money to the

main US political parties.

• There is also the American Sugar Alliance, which

lobbies for protection because farmers, its members,

benefit from it.

57

Sugar case

• Small foreign sugar producers do not have much

power to influence US trade policy.

• To consumers, the loss is only $8 per person

per year, so they don’t bother

• If US shifted to free trade, employment in sugar

industry would probably decline only by 3,000

workers, who would find new jobs.

58

Table: Protection to Maintain Jobs,

the United States

59

Table: Protection to Maintain Jobs,

the European Union

60

Figure 10.4 – Can an Import Barrier Be Better

Than Doing Nothing, and Is It the Best Policy?

61

Input-Output is essentially an accounting framework

Receipts Expenditures

Sales to Industries

Sales to Institutions

Exports

Purchases of goods and servicesLocalImported

Investment

Payroll

Taxes

ProfitsDistributedRetained

T - Account

Input-Output Analysis as an

Accounting Framework

62


Interindustry Transactions + Final Demands = Total Activity

Total Activity = f (Final Demand)

The economy is driven by consumption or final use

Industries contribute goods and services for final demand or

to those activities triggered by final consumption.

63

Input-Output Analysis – Another Example

I/O Tables - Transactions Transactions Table ($millions)

Purchasing Sectors

Processing

Sectors Agriculture Manufacturing Services

Final

Demand

Total

Output

Agriculture 10 6 2 18 36

Manufacturing 4 4 3 26 37

Services 6 2 1 35 44

Payments 16 25 38 0 79

Total Outlay 36 37 44 79 196

64

I/O Tables - Direct Requirements


Purchasing Sectors

Processing

Sectors Agriculture Manufacturing Services

Final

Demand

Total

Output

Agriculture .27778 .16216 .04545

Manufacturing .11111 .10811 .06818

Services .16667 .05405 .02273

Payments .44444 .67567 .86363

Total Outlay 1.0 1.0 1.0


65

Direct requirements in equation form:

X1 = 0.278 * X1 + 0.162 * X2 + 0.045 * X3 + Y1

X2 = 0.111 * X1 + 0.108 * X2 + 0.068 * X3 + Y2

X3 = 0.167 * X1 + 0.054 * X2 + 0.023 * X3 + Y3

X1 .278 .162 .045 X1 Y1

X2 = .111 .108 .068 * X2 + Y2

X3 .167 .54 .023 X3 Y3

X = A * X + Y


66

Subtract the direct requirements from both

sides of the equation:

X1 - 0.278 * X1 - 0.162 * X2 - 0.045 * X3 = Y1

X2 - 0.111 * X1 - 0.108 * X2 - 0.068 * X3 = Y2

X3 - 0.167 * X1 - 0.054 * X2 - 0.023 * X3 = Y3

X1 .278 .162 .045 X1 Y1

X2 - .111 .108 .068 * X2 = Y2

X3 .167 .54 .023 X3 Y3

X - A * X = Y


67

Collect terms:

(1 - 0 .2 7 8 ) * X 1 - 0 .1 6 2 * X 2 - 0 .0 4 5 * X 3 = Y 1

-0 .1 1 1 * X 1 + (1 -0 .1 0 8 ) * X 2 - 0 .0 6 8 * X 3 = Y 2

-0 .1 6 7 * X 1 - 0 .0 5 4 * X 2 + (1 -0 .0 2 3 ) * X 3 = Y 3

1 0 0 .2 7 8 .1 6 2 .0 4 5 X 1 Y 1

0 1 0 - .1 1 1 .1 0 8 .0 6 8 * X 2 = Y 2

0 0 1 .1 6 7 .5 4 .0 2 3 X 3 Y 3

(1 -A ) * X = Y

(1 -A ) -1 * (1 -A ) X = (1 -A ) -1 * Y

X = (1 -A ) -1 * Y


68

Predictive Model:

DTIO = (I-A)-1 * DFD


Same as the simple example we covered in Part I.

69

PART IV:

Input-Output Multipliers

70

What are Multipliers?

Multipliers measure total change throughout

the economy from one unit change for a

given sector.

71

Three Types of Multipliers

Output

Employment

Income

72

Three Levels of Multipliers

Type I Multipliers

Type II Multipliers

Type III Multipliers

73

Type I Multipliers

Include direct or initial spending

Include indirect spending or businesses

buying and selling to each other

The multiplier is direct plus indirect effect

divided by direct effect

74

Type II Multipliers

Includes Type I Multiplier effects

Plus household spending based on the

income earned from the direct and indirect

effects – the so called “induced effects”

75

Type III Multipliers

Type III Multipliers are modified Type II

multipliers.

Therefore, Type III Multipliers also include the

direct, indirect, and induced effects.

Type III Multipliers adjust Type II Multipliers

based on spending patterns amongst different

income groups.

76

Type I Multipliers include:

Direct Effects

Indirect (Business Spending) Effects

Type I Multipliers are derived from the

“Total Requirements Table”.

In math, this is: X = (1-A)-1 Y

77

The Leontief inverse of the direct requirements

table produces the table of total requirements.

Power Series:

(1 + A + A2 + A

3 + ...) = (1 - A)

-1

Limit of power series is Leontief inverse

Used in temporal studies

The more leakages the smaller the result


78

Sellin

g S

ectors

Purchasing Sectors

Agriculture Health Services

Agriculture 0.278 0.162 0.045

Health 0.111 0.108 0.068

Services 0.167 0.054 0.023

Final Payments 0.444 0.676 0.864

Total 1.000 1.000 1.000

Example: Direct Requirements Table

79

Sellin

g S

ectors

($ m

illion

)

Purchasing Sectors ($ million)

Agriculture Health Services

Agriculture 1.446 0.268 0.085

Health 0.199 1.163 0.090

Services 0.258 0.110 1.043

Total 1.903 1.541 1.218

Example: Total Requirements Table (Direct + Indirect Coefficients Table)

80

Explaining the Health Sector

Type I Multiplier

For a dollar change in final demand to

health sector, there will additional demand

on health services of 1.163, plus .268 from

agriculture, plus .11 from services, or a total

change of 1.541 in the regional economy.

81

Type II Multipliers include:

Direct Effects

Indirect (Businesses) Effects

Induced (Households) Effects

Type II Multipliers are derived from the “Total

Requirements Table with Households”.

82

Sellin

g S

ectors

($ m

illion

)

Purchasing Sectors ($ million)

Ag Health Services House- Final Total

holds Demands Output

Ag 10 6 2 2 16 36

Health 4 4 3 10 16 37

Services 6 2 1 7 28 44

Households 3 6 10 0 0 19

Final 13 19 28 0 0 60

Payments

Total Input 36 37 44 19 60 196

Example: Transactions Table with Households

83

Sellin

g S

ectors

Purchasing Sectors

Agriculture Health Services Households

Agriculture 1.536 0.369 0.197 0.429

Health 0.386 1.370 0.318 0.879

Services 0.388 0.256 1.203 0.619

Households 0.279 0.311 0.341 1.319

Total 2.589 2.307 2.059 3.245

Example: Total Requirements Table with Households

Explaining the Health Sector

Type II Multiplier

For a $1.00 change in final demand sales in the

local economy, the total direct, indirect and

induced impacts are $2.307

85

Multipliers

Direct requirements represent direct or initial

spending

Direct and indirect effects include the direct spending

plus the indirect spending or businesses buying and

selling to each other

Direct, indirect and induced effects include direct and

indirect plus household spending earned from direct

and indirect effects

86

Other Multipliers

• Employment Multipliers

Type I

Type II

Type III

• Income Multipliers

Type I

Type II

Type III

87

Example -

Type I Employment Multiplier

Agricultural Sector Type I Employment

Multiplier = 1.43

When the agricultural sector realizes a one employee

change, total employment in the study area changes by

1.43 jobs from direct and indirect linkages.

88

Example –

Type II Employment Multiplier

Agricultural Sector Type II Employment

Multiplier = 2.25

When the agricultural sector realizes a 1 employee

change, total employment in the study area changes by

2.25 jobs from direct, indirect and induced linkages.

89

Breakdown of

Type II Employment Multiplier -

Agricultural Sector

Direct Effects = 1.00

Indirect Effects = 0.43

Induced Effects = 0.82

Total = 2.25

90

Example –

Type I Income Multiplier

Agricultural Sector Type I Income Multiplier =

1.96

When the Agricultural Sector realizes a $1.00 change in

income, total income in the study area changes by $1.96

from direct and indirect linkages

91

Example -

Type II Income Multiplier

Agricultural Sector Type II Income Multiplier =

2.50

When the Agricultural Sector realizes a $1.00 change in

income, total income in the study area changes by $2.50

from direct, indirect and induced linkages

92

Breakdown of

Type II Income Multiplier -

Agricultural Sector

Direct Effects = $1.00

Indirect Effects = $0.96

Induced Effects = $0.54

Total = $2.50

93

Caution When Using Multipliers

Multiplier values include direct effects

Do not aggregate sector multipliers to

derive an aggregate multiplier

Be cautious of large multipliers

Be cautious in using a multiplier from

another study area

96

PART V:

Input-Output Example USA

97

A Regional Input-Output Model for the

US

Focuses on inter-industry transactions

Two suppliers: intermediate and primary (labor)

Two purchasers: intermediate and final

Composed of: Transaction table

Direct requirements table

Total requirements table

98

Transaction Table Start

Plastics Electricity Chemicals Autos Instruments Rubber

Other Local

Industries

$0.14 of auto

industry

spending on

plastics re-

enters:

$1 of additional

spending on

auto production

initiates

spending on:

$0.21 of auto

industry

spending on

other local

industries re-

enters:

Plastics $0.14

Electricity $0.05

$0.01

Chemicals $0.09

Autos $0.05

Instruments $0.11

Rubber $0.07

Other Local

Industries

$0.21

$0.04

Local Employees

$0.17

$0.02

$0.04

Leakage

$0.25

$0.02

$0.04

99



Other Local

Industries

$0.14 of auto

industry

spending on

plastics re-

enters:

$1 of additional

spending on

auto production

initiates

spending on:

$0.21 of auto

industry

spending on

other local

industries re-

enters:

Plastics $0.14

Electricity $0.05

$0.01

Chemicals $0.09

Autos $0.05

Instruments $0.11

Rubber $0.07

Other Local

Industries

$0.21

$0.04

Local Employees

$0.17

$0.02

$0.04

Leakage

$0.25

$0.03

$0.04

100



Other Local

Industries

$0.14 of auto

industry

spending on

plastics re-

enters:

$1 of additional

spending on

auto production

initiates

spending on:

$0.21 of auto

industry

spending on

other local

industries re-

enters:

Plastics $0.14

Electricity $0.05

$0.01

Chemicals $0.09

Autos $0.05

Instruments $0.11

Rubber $0.07

Other Local

Industries

$0.21

$0.04

Local Employees

$0.17

$0.02

$0.04

Leakage

$0.25

$0.03

$0.07

101

Transaction Table Sample


Intermediate Final Total

Suppliers Agriculture Manufacturing Services Purchasers Output



Services 25 10 5 20 60

Primary Suppliers

Households 60 10 40 110

Total Outlay 100 60 60 110 330

102


Derived from the transaction table

Shows inputs required from each supplier

by each intermediate purchaser.

“Direct coefficients” = each input purchase

in a column of the transaction table divided

by total purchases (column sum).

103

Transaction Table Sample


Intermediate Final Total

Suppliers Agriculture Manufacturing Services Purchasers Output



Services 25 10 5 20 60

Primary Suppliers

Households 60 10 40 110

Total Outlay 100 60 60 110 330

104


$1 of Output By

Requires Inputs

From Agriculture Manufacturing Services

Agriculture 0.10000 0.50000 0.08333

Manufacturing 0.05000 0.16667 0.16667

Services 0.25000 0.16667 0.08333

Households 0.60000 0.16667 0.66667

Total Outlay 1.00 1.00 1.00

Each column is the industry’s production function

105


“Spending Rounds”

Derived from the direct requirements table

and shows the total purchases of direct and

indirect inputs required throughout the

economy per unit of output sold to final

purchasers by each intermediate supplier.

106


Sales to Final

Purchasers To Agr. To Mfg To Serv. Total To Agr. To Mfg To Serv. Total

By Agriculture 200 (.1)(200) (.5)(100) (.08)(100) (.1)(78.33) (.5)(43.33) (.08)(75)

20 50 8.33 78.33 7.83 21.67 6.25 35.75

By Manufacturing 100 (.05)(200) (.17)(100) (.17)(100) (.05)(78.33) (.17)(43.33) (.17)(75)

10.00 16.67 16.67 43.33 3.92 7.22 12.50 23.64

By Services 100 (.25)(200) (.17)(100) (.08)(100) (.25)(78.33) (.17)(43.33) (.08)(75)

50.00 16.67 8.33 75.00 19.58 7.22 6.25 33.06

By Households (.6)(200) (.17)(100) (.67)(100) (.6)(78.33) (.17)(43.33) (.67)(75)

120.00 16.67 66.67 203.33 47.00 7.22 50.00 104.22

Totals - Indirect Rounds 196.67

By all Suppliers 400 400.00

Second RoundSales as Direct Inputs

107

Impacts Broken Down

Direct impacts – the initial injection of new

economic activity, i.e., a new mfg plant

locates in a state.

Indirect impacts – the sum of inter-industry

purchases through all the rounds of

purchasing

Induced impacts – the sum of all impacts

associated with employee expenditures

108

Multiplier

Output multiplier

Income multiplier

Employment multiplier

Direct + Indirect + Induced

Direct

109

The Power of IO Models

IO analysis is a popular and powerful analytical tool.

“The chief value of regional input-output analysis is in

its descriptive analytical power.” (Bendavid-Val, p.113)

“As a descriptive tool, input-output tables:

-present an enormous quantity of information in a

concise, orderly, and easily understood fashion;

-provide a comprehensive picture of the interindustry

structure of the regional economy;

-point up the strategic importance of various

industries and sectors;

-highlight possible opportunities for strengthening

regional income and employment multiplication.”

(Bendavid-Val, p.113)

110

The Problems with IO Analysis Practical Issues

Data needs and complexity: IO models are tremendously complex and very

data hungry. This typically places these models in the hands of experts.

Theoretical Issues

Time/Data issues: Usually a single year’s data are used to develop the Total

Requirements Table. But 1) purchases may actually reflect a longer term

investment and 2) short term trends may impact the data.

Stability of the technical coefficients over time: Technology changes, prices

change, and demand changes, all affecting the coefficients in the Tot Reqs

Table. This can impact the results if the coefficients are “out of date”.

IO assumes a linear relationship between increasing demand for inputs

and outputs: This assumes away 1) externalities and 2) increasing/

decreasing returns to scale.

Industrial categorization: IO models still assume that each industry 1) has

a single, homogeneous production function and 2) each produces one product. These assumptions do not reflect the real economy very well.

111

Thank you.

112

APPENDIX 1:

Mathematical Exercises on IO Analysis with Matrices

113

Appendix I. Input-output analysis – an application of matrices.

Learning objectives. By the end of this lecture you should:

– Know more about input – output analysis

– Understand how to calculate input requirements given an output

requirement.

– Understand how to check for the productiveness of an input-output

system.

1. Introduction: Inputs and outputs.

• Input-output analysis was developed in the early years after the

introduction of national accounting systems

• It builds on the fact that any one sector will use inputs from many other

sectors of the economy. They in turn will use inputs from many sectors.

– E.g. the IT industry requires electricity, but electrical generation

uses computers to control its output.

• It’s a method of planning resource use.

• It’s also used to calculate employment and output multiplier effects.

114

1. Use data on which sector buys from which

2. E.g. consider this set of national accounts

Final demands are demands made by the household or government sector for final consumption; final payments are payments made to owners of the final inputs (labour, capital, land etc.)

2. Constructing the input coefficients matrix

Buying

sector

IT

goods

Services Transport Final

demands

Total

Selling

sector

IT goods £100m

£400m £200m £200m £900m

Services £100m £100m £400m £800m £1400m

Transport £100m £200m £100m £200m £600m

Final

payments

£100m £300m £300m £700m

Total 400 1000 £1000m 1200 £3600

115

1. Turn into per £ of output:

2. E.g.

2. Constructing the input coefficients matrix

Buying sector IT goods Services Transport

Selling sector

IT goods 0.25

0.40 0.2

Services 0.25 0.10 0.40

Transport 0.25 0.20 0.1

Total 0.75 0.7 0.7

116

3. Using the input coefficients matrix

Use the vector x to denote inputs and d to denote final demand. Use A for

the input coefficients matrix:

So aij is the value of input i required to produce 1 unit of value of good j.

Note that if input i is not used in the production of good j then aij=0.

Otherwise aij > 0. A is therefore a positive matrix.

Note also that for each good inputs are used in fixed proportions – there is

no possibility of substitution. The underlying assumption is that the

technology is Leontief (named after the man who invented input-output

analysis).

nnn

n

aa

aa

A

1

111

Input 1

Input 2

ai1

ai2

117

3. Using the matrix

The matrix A can be used either:

1. to calculate inputs required given a vector of final demands or

2. To calculate final outputs given a vector of available inputs.

aij is the value of good i required as input to produce 1 unit of good j.

So xj aij is the amount of good i required to produce xj units of good j.

It follows that the total demand for good i is:

Thus:

Or x = Ax + d

ininii dxaxax 11

nnnnn

n

n d

d

x

x

aa

aa

x

x

11

1

1111

118

3. Using the matrix

x = Ax + d can be rewritten as Ix = Ax + d where I is the identity matrix.

So (I-A)x = d:

Example. Suppose

and 1 unit of good 1, 1 unit of good 2 and 1 unit of good 3 are available.

What is the resulting final demand?

nnnnn

n

d

d

x

x

aa

aa

11

1

111

1

1

1.02.025.0

4.01.025.0

2.04.025.0

A

55.0

25.0

15.0

1

1

1

9.02.025.0

40.09.025.0

2.04.075.0

)( xAI

119

Exercise

Example. Suppose

and 2 units of good 1 and 1 unit of good 2 are available. What is the

resulting final demand for these two goods?

1.03.0

1.02.0A

120

4. Finding inputs given final demand

If (I-A)x = d then provided det (I-A) ≠ 0 then we can invert the matrix to find

x:

x = (I-A)-1d

Example. Suppose

Then

Det (I-A) = 0.75(0.81-0.08)+0.4(-0.225-0.1)-0.2(0.05+0.225)=0.3715 so the

matrix is invertible.

Suppose final consumption is 1 unit of good 1, 2 units of good 2 and 1 unit

of good 3. What is x?

9.02.025.0

40.09.025.0

2.04.075.0

)( AI

1.02.025.0

4.01.025.0

2.04.025.0

A

121

4. Finding inputs given final demand

x = (I-A)-1d

Then

N.b. this inverse is done using the minverse command in excel, so it’s only

approximate.

So

Note that this multiplication is done using the mmult command in excel, so

it’s only approximate.

59.169.076.0

97.072.190.0

94.010.101.2

)( 1AI

72.3

31.5

16.5

1

2

1

59.169.076.0

97.072.190.0

94.010.101.2

)( 1dAIx

122

Exercise

Suppose

What is (I-A)-1

What is x if

2.01.0

0.02.0A

2

1d

123

5. Final or primary inputs

Final inputs or primary inputs are those which are not produced goods

within the economic system being studied. They may include:

• Labour

• Land

• Capital (sometimes)

• Imports (sometimes)

Final inputs receive the final payments.

Obviously if a vector of final demand is to be feasible the final inputs must

be sufficient.

124


Since final inputs receive the final payments the requirement for primary

inputs is given by 1 – sum of the column entries in the input matrix

In the example, £1 of IT goods needs £0.25 input of primary inputs.

The primary inputs coefficient vector, l, is the vector of these values:


Selling sector

IT goods 0.25 0.40 0.2

Services 0.25 0.10 0.40

Transport 0.25 0.20 0.1

Total 0.75 0.7 0.7

Final inputs 0.25 0.3 0.3

30.0

30.0

25.0

l

125


So the total demand for primary inputs is then

In the example:

For instance if the primary input is labour then this means that £4 of labour

is required to produce a final demand consisting of £1 of good 1, £2 of

good 2 and £1 of good 3.

Feasibility then consists of comparing this demand for the primary inputs to

the available supply, L.

If then a vector of final demands is feasible.

Here if L = 3, then the vector of final demands was not feasible.

dAIlxl 1)(

4

72.3

31.5

16.5

3.03.025.0)( 1

dAIl

LdAIl 1)(

126

6. Summary.

• 4 definitions learnt:

– Input coefficient matrix

– Final demands

– Final payments

– Primary inputs

• 4 skills you should be able to do:

– Write down an input coefficient matrix given a table of input values.

– Determine input demand given a vector of final demands

– Determine final demands given a vector of inputs

– Find primary input demands and check their feasibility

127

Lecture 14. Input-output analysis II.

Learning objectives. By the end of this lecture you should:

– Know more about input – output analysis

– Understand how to check for the productiveness of an input-output system.

1. Introduction: Inputs and outputs.

• In the last lecture we learnt the basics of input-output analysis.

• A key question is whether a given input coefficients matrix makes sense, meaning:

– Given the input coefficient matrix and provided there are sufficient primary inputs, can any pattern of final demands be produced?

– Example. OK Computers PLC buys PCs from 3 suppliers. From the first it keeps the keyboard and throws everything else away. From the second it keeps the monitor, throwing everything else away and from the third it throws away the monitor and keyboard. So from 3 computers it produces 1 new one.

128

1. A: I-A:

2. So (I-A)-1 =

3. If then x = (I-A)-1d =

4. In other words, to meet final demand for 1 unit of the first good, 1.33

units of that good must be produced along with 0.46 units of good 2

and 0.53 units of good 3.

2. Example

0.1 0.2 0.2

0.2 0.3 0.1

0.25 0.2 0.2

1.33 0.49 0.39

0.46 1.65 0.32

0.53 0.57 1.45

0

0

1

d

53.0

46.0

33.1

0.9 -0.2 -0.2

-0.2 0.7 -0.1

-0.25 -0.2 0.8

129

Another way to think about this:

To meet final demand d we require:

d the final demand

+ Ad the direct intermediate inputs

+ A(Ad) the inputs required for the direct intermediate inputs

+ A(A2d) the inputs required…

+ …

Or x = d + Ad + A2d + A3d + ….= (I+A+A2+A3+…)d

Question on feasibility:

Given any final demand vector d, is there an input vector x that will

produce d?

1. Obviously to be feasible no element of x can be negative

2. And no element can be infinite

3. Meeting demand

130

Given that,

• x = d + Ad + A2d + A3d + ….= (I+A+A2+A3+…)d and

• All the elements of A are non-negative (so that all the elements of An

must be positive), then

• It’s clear that x won’t be negative. But can it be infinite?

Define S = (I+A+A2+A3+…)

Define Sn = (I+A+A2+A3+…An)

In other words S is the limit of Sn as n →∞.

Note that ASn = A+A2+A3+…An+1

So Sn – ASn = (I+A+A2+A3+…An) – (A+A2+A3+…An+1)

= I - An+1

Or

(I-A)Sn = I - An+1

So if (I-A)S = I or S = (I-A)-1 and x exists.

3. Meeting demand

0lim

n

nA

131

The Hawkins-Simons conditions are conditions on A which guarantee that

given any final demand vector, d, there is an input vector, x, which will

produce d.

There are different, but equivalent statements of the conditions. We shall

state 3 and consider the first 2:

1.

2. The principal minors** of (I-A) are all positive.

3. The dominant eigenvalue is less than 1.

(don’t know what an eigenvalue is? Given a matrix A it’s a solution, λ

to the equation . The dominant eigenvalue is the one

with the largest absolute value)

** we are going to define this term on the next slide

If the conditions are satisfied then the input-output system is said to be

productive.

4. Hawkins-Simons conditions

0lim

n

nA

0 AI

132

1.

This is the condition we derived earlier. It’s simple, but may be hard to

calculate.

2. The principal minors** of (I-A) are all positive.

1. Given an nxn matrix, A, a principal matrix is found from A by

deleting k rows (e.g. rows 2, 5 and 7) and the same k columns (so

columns 2, 5 and 7). 0≤ k ≤ n-1. k is called the order of the

principal matrix.

2. The principal minor is the determinant of the relevant principal

matrix.

3. So for a 3x3 matrix, there are:

1. 1 0-order principal matrix (A itself)

2. 3 first order principal matrices

3. 3 second order principal matrices.

So you would need to check 7 determinants.

4. Hawkins-Simons conditions

0lim

n

nA

133

1. Suppose A = Are the conditions satisfied?

2. I-A is

3. The Principal matrices are I-A itself and,

4. The relevant determinants are:

0.352, 0.52, 0.59, 0.54, 0.8, 0.7, 0.8, so the conditions are satisfied.

Note how when we eliminate n rows and columns we end up with the

diagonal elements of A.

4. Hawkins-Simons conditions -example

0.2 0.2 0.2

0.2 0.3 0.1

0.25 0.2 0.2

0.8 -0.2 -0.2

-0.2 0.7 -0.1

-0.25 -0.2 0.8

0.8 0.7 0.8

0.8 -0.2 -0.2

-0.2 0.7 -0.1

-0.25 -0.2 0.8

0.8 -0.2 -0.2

-0.2 0.7 -0.1

-0.25 -0.2 0.8

0.8 -0.2 -0.2

-0.2 0.7 -0.1

-0.25 -0.2 0.8

134

1. Suppose A =

Find the principal matrices (no need to calculate their determinants).

Note: principal minors will resurface in the next topic.

4. Hawkins-Simons conditions -exercise

0.2 0 0.2

0.1 0.4 0.3

0.3 0.1 0.4

135

1. Suppose two goods, computers and software. To produce the

computer 0.1 unit of software is required and to produce 1 unit of

software, 0.3 units of computer and 0.2 units of software are needed.

2. A =

I-A =

Principal minors are, -2, 0.8 and -1.63. Obviously not all positive.

Note the basic point here: if it takes more than one unit of a good to

produce that good, then the Hawkins-Simons conditions can never be

satisfied.

4. Hawkins-Simons conditions –OK computers example

3 0.3

0.1 0.2

-2 -0.3

-0.1 0.8

136

Often the number of jobs apparently lost as the result of a business closing

or the number of jobs created as the result of new business opening

seems far in excess of the number of jobs actually with the specific

company.

Input output analysis can be used to calculate the knock on effects of

changes in employment.

5. Employment multipliers

137

Suppose the only primary input is labour. Recall that,

Note first that this labour demand is in value terms

Consider a change in final demand of Δd, then the change in the value of

labour demand will simply be Δd (because of the circular flow of

payment within the economy and the fact that we assumed that there is

only one input).

It is still useful to break this down into sectors using:


dAIlxldemandlabour 1)(

dAIlL DD 1)(

138

How we calculated l (previous lecture)

Recall final inputs receive the final payments the requirement for primary

inputs is given by 1 – sum of the column entries in the input matrix

In the example, £1 of IT goods needs £0.25 input of primary inputs.

The primary inputs coefficient vector, l, is the vector of these values:


Selling sector

IT goods 0.25 0.40 0.2

Services 0.25 0.10 0.40

Transport 0.25 0.20 0.1

Total 0.75 0.7 0.7

Final inputs 0.25 0.3 0.3

30.0

30.0

25.0

l

139

Example. Suppose d1 falls by one unit, what happens to the value of

employment?


DD

0

0

1

59.169.076.0

97.072.190.0

94.010.101.2

3.03.025.0)( 1 dAIlL

23.027.05.0

76.0

9.0

01.2

3.03.025.0

140

Now l‘1 = 0.25, so from a 1 unit drop in demand we get

• a 0.25 direct drop in the value of employment in sector 1

• a further 0.25 indirect drop in the value of employment in sector 1

• a 0.27 indirect drop in sector 2

• a 0.23 indirect drop in sector 3.

• The employment multiplier is the ratio of the total change in

employment to the direct drop.

• The multiplier is therefore

• I.e. for every one job lost due to the immediate effect of the drop in

demand, there are 3 jobs lost indirectly.


23.027.05.0

76.0

9.0

01.2

3.03.025.0

425.0

23.027.025.025.0

141

Conclusion.

6. Exercise.

identify the direct and indirect effects on employment if Δd2 = 2

what is the employment multiplier for d2?

142

Conclusion.

7. Summary

• 4 definitions learnt:

– Hawkins-Simons conditions

– Principal minors

• 4 skills you should be able to do:

– Identify the Principal minors of a square matrix

– Check that the Hawkins Simons are satisfied

– Calculate employment multipliers.

143

APPENDIX 2:

Derivation of the multipliers

from transaction table to direct requirements table

via Leontief Inverse Matrix total requirements table

144

Input-Output table

tota

l

95

300

240ag

ricu

ltu

re

ind

ust

ry

serv

ices

agriculture 10 30 5

industry 35 70 50

services 15 50 70

imports 15 75 15

wages and taxes 10 25 80

profits 10 50 20

total 95 300 240ex

po

rts

con

sum

pti

on

inv

estm

ents

20 30 0

70 40 35

30 70 5agriculture

industry

services

Value added

Final demand

145

Technological matrix; Cost

structure

agri

cult

ure

ind

ust

ry

serv

ices

agriculture 10 30 5

industry 35 70 50

services 15 50 70

agri

cult

ure

indust

ry

serv

ices

0.11 0.10 0.02

0.37 0.23 0.21

0.16 0.17 0.29

0.11 0.08 0.33

0.11 0.17 0.08

1.00 1.00 1.00

0.16 0.25 0.06imports 15 75 15

wages and taxes 10 25 80

profits 10 50 20

total 95 300 240

A

146

Input-Output model

x = A x + y x – A x = y

(I - A) x = y

x = (I - A)-1 y

with

x vector of total production

A technological matrix

I unit matrix

y vector of final demand

147

Leontief inverse

• (I - A)-1 ; multipliers

– Total inputs required for one unit of final

demand for all sectors • A

– First order (direct) inputs for one unit of

production

• A + A2 + A3 + …

– First and higher order inputs for one unit of

production

• I + A + A2 + A3 + … = (I - A)-1

– Total inputs required for one unit of final

demand for all sectors

148

Leontief inverse matrix:

multipliers A

agri

cult

ure

ind

ust

ry

serv

ices

agriculture 0.11 0.10 0.02

industry 0.37 0.23 0.21

services 0.16 0.17 0.29

(I - A)-1

agri

cult

ure

indust

ry

serv

ices

1.21 0.18 0.09

0.70 1.50 0.46

0.43 0.39 1.54

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